The Sense of Place:Grid Cells in the Brain and the Transcendental Number e

Xue-Xin Wei,1 Jason Prentice,2,5 Vijay Balasubramanian2,3,4∗

1 Department of Psychology, University of Pennsylvania, Philadelphia, PA 19104, USA2 Department of Physics, University of Pennsylvania, Philadelphia, PA 19104, USA

3 Department of Neuroscience, University of Pennsylvania, Philadelphia, PA 19104, USA4 Laboratoire de Physique Theorique, Ecole Normale Superieure, 75005 Paris, France5 Princeton Neuroscience Institute, Princeton University, Princeton, NJ 08544, USA

∗Address correspondence to: vijay@physics.upenn.edu.

Grid cells in the brain respond when an animal occupies a periodic lattice of“grid fields” during spatial navigation. The grid scale varies along the dorso-ventral axis of the entorhinal cortex. We propose that the grid system minimizesthe number of neurons required to encode location with a given resolution. Wederive several predictions that match recent experiments: (i) grid scales follow ageometric progression, (ii) the ratio between adjacent grid scales is

√e for ide-

alized neurons, and robustly lies in the range 1.4-1.7 for realistic neurons, (iii)the scale ratio varies modestly within and between animals, (iv) the ratio betweengrid scale and individual grid field widths at that scale also lies in this range, (v)grid fields lie on a triangular lattice. The theory also predicts the optimal grids inone and three dimensions, and the total number of discrete scales.

Introduction

How does the brain represent space? Tolman (1) suggested that the brain must have an explicit neu-ral representation of physical space, a cognitive map, that supports higher brain functions such asnavigation and path planning. The discovery of place cells in the rat hippocampus (2,3) suggestedone potential locus for this map. Place cells have spatially localized firing fields which reorganizedramatically when the environment changes (4). Another potential locus for the cognitive mapof space has been uncovered in the main input to hippocampus, a structure known as the medialentorhinal cortex (MEC) (5, 6). When rats freely explore a two dimensional open environment,individual “grid cells” in the MEC display spatial firing fields that form a periodic triangular grid

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which tiles space (Fig. 1A). It is believed that grid fields provide relatively rigid coordinates onspace based partly on self-motion and partly on environmental cues (7). Locally within the MEC,grid cells share the same orientation and periodicity, but vary randomly in phase (6). The scale ofgrid fields varies systematically along the dorso-ventral axis of the MEC (Fig. 1A) (6, 8).

How does the grid system represent spatial location and what function does the striking trian-gular lattice organization and systematic variation in grid scale serve? Here, we begin by assumingthat grid cell scales are organized into discrete modules (8), and propose that the grid system fol-lows a principle of economy by minimizing the number of neurons required to achieve a givenspatial resolution. Our hypothesis, together with general assumptions about tuning curve shapeand decoding mechanism, predicts a geometric progression of grid scales. The theory furtherdetermines the mean ratio between scales, explains the triangular lattice structure of grid cell fir-ing maps, and makes several additional predictions that can be subjected to direct experimentaltest. For example, the theory predicts that the ratio of adjacent grid scales will be modestly vari-able within and between animals with a mean in the range 1.4 − 1.7 depending on the assumeddecoding mechanism used by the brain. This prediction is quantitatively supported by recent ex-periments (8,13). In a simple decoding scheme, the scale ratio in an n-dimensional environment ispredicted to be close to n

√e. We also estimate the total number of scales providing the spatial reso-

lution necessary to support navigation over typical behavioral distances, and show that it comparesfavorably with estimates from recent experimental measurements (8).

Results

General grid coding in one dimension

Consider a one dimensional grid system that develops when an animal runs on a linear track.Suppose that grid fields develop at a discrete set of periodicities λ1 > λ2 > · · · > λm (Fig. 1A).We will refer to the population of grid cells sharing each periodicity λi as one module. It willprove convenient to define “scale factors” ri ≡ λi

λi+1. Here λ1 could be the length of the entire

track and we do not assume any further relation between the λi, such as a common scale ratio (i.e.,in general r1 6= r2 6= · · · 6= rm−1). Now let the widths of grid fields in each module be denotedl1, l2, · · · lm. Within any module, grid cells have a variety of spatial phases so that at least one cellmay respond at any physical location (Fig. 1D). To give uniform coverage of space, the numberof grid cells ni at scale i should be proportional to λi/li – thus we write ni = dλi/li in terms of a“coverage factor” d that represents the number of grid fields overlapping each point in space. Weassume that d is the same at each scale. In terms of these parameters, the total number of grid cellsis N =

∑mi=1 ni =

∑mi=1 d

λili

.

Grid cells with smaller scales provide more local spatial information than those with largerscales, owing to their smaller li. However, this increased resolution comes at a cost: the smallerperiodicity λi of these cells leads to increased ambiguity (Fig. 1C,E, Fig. 2A-D). In this paper, westudy coding schemes in which information from grid cells with larger scales is used to resolvethis ambiguity in the smaller scales, while the smaller scales provide improved local resolution(Fig. 1E). In such a system, resolution may thus be improved by increasing the total number of

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Figure 1: Representing place in the grid system. (A) Grid cells (small triangles) in the medialentorhinal cortex (MEC) respond when the animal is in a triangular lattice of physical locations(red circles; sometimes also called a “hexagonal lattice”) (5, 6). The scale of periodicity (the“grid scale”, λi) and the size of the regions evoking a response (the “grid field width”, li) varysystematically along the dorso-ventral axis of the MEC (6). (B) A simplified binary grid schemefor encoding location along a linear track. At each scale (λi) there are two grid cells (red vs.blue firing fields). The periodicity and grid field widths are halved at each successive scale. (C)Decoding is ambiguous if the grid field width at scale i exceeds the grid periodicity at scale i+ 1.E.g., if the grid fields marked in red respond at scales i and i + 1, the animal might be in eitherof the two marked locations. (D) We extend the binary code of panel B to the more realistic caseof populations of noisy neurons with overlapping tuning curves. (E) The relationship betweengrid periodicity, λi, and grid field width, li. In the winner-take-all case, decoded position will beambiguous unless li ≤ λi+1, analogously to the situation depicted in panel C.

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Figure 2: (A-D) Trade-off between precision and ambiguity in the Bayesian decoder. (A) In-formation about position given the responses of all grid cells at scales smaller than module i issummarized by the posterior Qi−1(x) (black curve), and the uncertainty in position is given bythe standard deviation δi−1. Grid cells in module i contribute the periodic posterior Pi(x) (greencurve). (B) The updated posterior combining module i with all larger-scale modules is given bythe product Qi(x) ∼ Pi(x)Qi−1(x), and has the reduced uncertainty δi. (C) Precision is improvedby increasing the scale factor, thereby narrowing the peaks of Pi(x). However, the periodicityshrinks as well, increasing ambiguity. (D) Posterior Qi(x) given by combining the modules shownin C. Ambiguity from the secondary peaks leads to an overall uncertainty δi larger than in B, de-spite the improved precision from the narrower central peak. There is thus an optimal scale factorsomewhere between that in A, B and in C, D. (E) The optimal ratio r between adjacent scales in ahierarchical grid system in one dimension for a simple winner-take-all decoding model (blue curve,WTA) and a Bayesian decoder (red curve). Here Nr is the number of neurons required to representspace with resolution R given a scaling ratio r, and Nmin is the number of neurons required at theoptimum. In both decoding models, the ratio Nr/Nmin is independent of resolution, R. For thewinner-take-all model, Nr ∝ r/ ln r, as derived in the main text, and the curve for the Bayesianmodel is derived numerically as described in Supplemental Sec. 5. The winner-take-all modelpredicts that the minimum number of neurons is achieved for r = e ≈ 2.7, while the Bayesiandecoder predicts r ≈ 2.3. The minima of the two curves lie within each others’ shallow basins. (F)Same as E, but in two dimensions with a triangular grid. The winner-take-all curve in this case isNr ∝ r2/ ln(r2) (see main text), and the minima occur at r =

√e ≈ 1.65 for winner-take-all and

r ≈ 1.44 for the Bayesian case. The shallowness of the basins around these minima predicts thatsome variability of adjacent scale ratios is tolerable, both within and between animals.

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modules m. Alternatively, the field widths li may be made smaller relative to the periodicities λi;however, this necessitates using more neurons at each scale in order to maintain the same coveraged. Improving resolution by either mode therefore requires additional neurons. An efficient gridsystem will minimize the number of grid cells providing a fixed resolutionR; we shall demonstratehow the parameters of the grid system, ri, li/λi, and m, should be chosen to achieve this optimalcoding. We will characterize efficient grid systems in the context of two decoding methods atextremes of complexity.

We first consider a decoder which consider the animal as localized within the grid field of themost responsive cell in each module (9, 10). Such a “winner-take-all” scheme is at one extreme ofdecoding complexity and could be easily implemented by neural circuits. Any decoder will have tothreshold grid cell responses at the background noise level, so that the firing fields are effectivelycompact (Fig. 1D). Grid cell recordings suggest that the firing fields are, indeed, compact (6). Theuncertainty in the animal’s location at grid scale i is given by the grid field width li. The smallestscale that can be resolved in this way is lm, we therefore define the resolution of the grid systemas the ratio of the largest to the smallest scale, R1 = λ1/lm. In terms of scale factors ri ≡ λi

λi+1, we

can write the resolution asR1 =∏m

i=1 ri, where we also defined rm ≡ λmlm

. Unambiguous decodingrequires that li ≤ λi+1 (Fig. 1C,E), or, equivalently, λi

li≥ ri. To minimize N = d

∑i λi/li, all the

λili

should be as small as possible; so this fixes λili

= ri. Thus we are reduced to minimizing the sumN = d

∑mi=1 ri over the parameters ri, while fixing the product R =

∏i ri. Because this problem

is symmetric under permutation of the indices i, the optimal ri turn out to all be equal, allowing usto set ri = r (Supplementary Material). Our optimization principle thus predicts a common scaleratio, giving a geometric progression of grid periodicities. The constraint on resolution then givesm = logr R, so that we seek to minimize N(r) = d r logr R with respect to r: the solution is r = e(Fig. 2E; details in Supplementary Material). Therefore, for each scale i, λi = e λi+1 and λi = e li.Here we treated N and m as continuous variables – treating them as integers throughout leads tothe same result through a more involved argument (Supplementary Material). The coverage factord and the resolution R do not appear in the optimal ratio of scales.

The brain might implement the simple decoding scheme above via a winner-take-all mecha-nism (9–11). But the brain is also capable of implementing far more complex decoders. Hence,we also consider a Bayesian decoding scheme that optimally combines information from all gridmodules. In such a setting, an ideal decoder should construct the posterior probability distributionof the animal’s location given the noisy responses of all grid cells. The population response at eachscale i will give rise to a posterior over location P (x|i), which will have the same periodicity λias the individual grid cells’ firing rates (Fig. 2A). The posterior given all m scales, Qm(x), will begiven by the product Qm(x) = N Πm

i=1P (x|i), assuming independent response noise across scales(Fig. 2B). Here N is a normalization factor. The animal’s overall uncertainty about its positionwill then be related to the standard deviation δm of Qm(x), we therefore quantify resolution asR = λ1/δm. δm, and therefore R, will be a function of all the grid parameters (Supplementary Ma-terial). In this framework, ambiguity from too-small periodicity λi decreases resolution, as doesimprecision from too-large field width li. We thus need not impose an a priori constraint on theminimum value of λi, as we did in the winner-take-all case: minimizing neuron number while fix-ing resolution automatically resolves the tradeoff between precision and ambiguity (Fig. 2A-D). Tocalculate the resolution explicitly, we note that when the coverage factor d is very large, the distri-

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butions P (x|i) will be well-approximated by periodic arrays of Gaussians (even though individualtuning curves need not be Gaussian). We can then minimize the neuron number, fixing resolution,to obtain the optimal scale factor r ≈ 2.3: slightly smaller than, but close to the winner-take-allvalue, e (Fig. 2E; details in Supplementary Material). As before, the optimal scale factors are allequal so we again predict a geometric progression of scales.

It is apparent from Fig. 2E that the minima for both the Bayesian decoder and the winner-take-all decoder are shallow, so that the scaling ratio r may lie anywhere within a basin aroundthe optimum at the cost of a small number of additional neurons. Even though our two decodingstrategies lie at extremes of complexity (one relying just on the most active cell at each scale andanother optimally pooling information in the grid population) their respective “optimal intervals”substantially overlap. That these two very different models make overlapping predictions suggeststhat our theory is robust to variations in the detailed shape of grid cells’ grid fields and the precisedecoding model used to read their responses. Moreover, such considerations also suggest that thesecoding schemes have the capacity to tolerate developmental noise: different animals could developgrid systems with slightly different scaling ratios, without suffering a large loss in efficiency.

General grid coding in two dimensions

How do these results extend to two dimensions? Let λi be the distance between neighboringpeaks of grid fields of width li (Fig. 1A). Assume in addition that a given cell responds on alattice whose vertices are located at the points λi(nu +mv), where n,m are integers and u,v arelinearly independent vectors generating the lattice (Fig. 3B). We may take u to have unit length(|u| = 1) without loss of generality, however |v| 6= 1 in general. It will prove convenient to denotethe components of v parallel and perpendicular to u by v‖ and v⊥, respectively (Fig. 3B). Thetwo numbers v‖, v⊥ quantify the geometry of the grid and are additional parameters that we mayoptimize over: this is a primary difference from the one-dimensional case. We will assume thatv‖ and v⊥ are independent of scale; this still allows for relative rotation between grids at differentscales.

At each scale, grid cells have different phases so that at least one cell responds at each physicallocation. The minimal number of phases required to cover space is computed by dividing the areaof the unit cell of the grid (λ2

i ||u × v|| = λ2i |v⊥|) by the area of the grid field. As in the one-

dimensional case, we define a coverage factor d as the number of neurons covering each point inspace, giving for the total number of neurons N = d|v⊥|

∑i(λi/li)

2.

As before, consider a simple model where grid fields lie completely within compact regionsand assume a decoder which selects the most activated cell (9–11). In such a model, each scale iserves to localize the animal within a circle of diameter li. The spatial resolution is summarizedby the square of the ratio of the largest scale λ1 to the smallest scale lm: R2 = (λ1/lm)2. In termsof the scale factors ri = λi/λi+1 we write R2 =

∏mi=1 r

2i , where we also define rm = λm/lm. To

decode the position of an animal unambiguously, each cell at scale i should have at most one gridfield within a region of diameter li−1. Since the nearest firing fields lie at a distance λi along thethree grid axes u,v, and u−v, we require min(|v|, |u−v|, 1)·λi ≥ li−1 in order to avoid ambiguity(Fig. 3C). To minimize N we must make λi−1/li−1 = ri−1λi/li−1 as small as possible, so that λi =li−1, which is only possible if |v| ≥ 1, |u− v| ≥ 1. We then have N = d|v⊥|

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2i . We now seek

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Figure 3: (A) Two dimensional analog of a grid scheme with circular firing fields. (B) A generaltwo-dimensional lattice may be parameterized by two vectors u and v and a periodicity parameterλi. We take u to be a unit vector, so that the spacing between peaks along the u direction isλi, and denote the two components of v by v‖, v⊥. The blue-bordered region is a fundamentaldomain of the lattice, the largest spatial region that may be unambiguously represented. (C) Thetwo dimensional analog of the ambiguity in Fig. 1C, E for the winner-take-all decoder. If thegrid fields in scale i are too close to each other relative to the size of the grid field of scale i − 1(i.e. li−1), the animal might be in one of several locations. (D) Contour plot of normalized neuronnumberN/Nmin in the Bayesian decoder, as a function of the grid geometry parameters v⊥, v‖ afterminimizing over the scale factors for fixed resolution R. As in Fig. 2E,F, the normalized neuronnumber is independent of R. The spacing between contours is 0.01, and the asterisk labels theminimum at v‖ = 1/2, v⊥ =

√3/2; this corresponds to the triangular lattice.

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parameters v‖, v⊥, ri that minimize N while fixing the resolution R2. Since R2 does not dependon the geometric parameters v‖, v⊥, we may determine these parameters by simply minimizingN , which is equivalent to minimizing |v⊥| subject to the constraints |v| ≥ 1, |u − v| ≥ 1. Thisoptimization picks out the triangular lattice with v⊥ =

√3/2, v‖ = 1/2. Note that this formulation

is mathematically analogous to the optimal sphere-packing problem, for which the solution intwo dimensions is also the triangular lattice (22). As for the scale factors ri, the optimizationproblem is mathematically the same as in one dimension if we formally set ri ≡ r2

i . This givesthe optimal ratio r2

i = e for all i (Fig. 2F). We conclude that in two dimensions, the optimal ratioof neighboring grid periodicities is

√e ≈ 1.65 for the simple winner-take-all decoding model, and

the optimal lattice is triangular.

The Bayesian decoding model can also be extended to two dimensions with the posterior distri-butions P (x|i) becoming sums of Gaussians with peaks on the two-dimensional lattice. In analogywith the one-dimensional case, we then derive a formula for the resolution R2 = λ1/δm in termsof the standard deviation δm of the posterior given all scales. δm may be explicitly calculated asa function of the scale factors ri and the geometric factors v‖, v⊥, and the minimization of neu-ron number may then be carried out numerically (Supplementary Material). In this approach theoptimal scale factor turns out to be ri ≈ 1.4 (Fig. 2F), and the optimal lattice is again triangular(Fig. 3D).

Once again, the optimal scale factors in both decoding approaches lie within overlapping shal-low basins, indicating that our proposal is robust to variations in grid field shape and to the precisedecoding algorithm (Fig. 2F). In two dimensions, the required neuron number will be no morethan 5% of the minimum if the scale factor is within (1.43, 1.96) for the winner-take-all modeland (1.28, 1.66) for the Bayesian model. These “optimal intervals” are narrower than in the one-dimensional case, and have substantial overlap.

The fact that both of our decoding models predicted the triangular lattice as optimal is a con-sequence of the fact that they share a very general symmetry. The resolution formula in bothproblems is invariant under a common rotation and a common rescaling of all firing rate maps.The neuron number shares this symmetry, as well. The rotation invariance implies that the reso-lution only depends on grid geometry through v⊥, v‖, and the rescaling invariance implies that itonly depends on λi, li through the dimensionless ratios ri, λi/li. However, even after restrictingthe parameters in this way, the rotation- and rescaling-invariance has a nontrivial consequence.The transformation v⊥ → −v⊥/|v|2, v‖ → v‖/|v|2, li → li/|v| can be seen to be equivalent toa rotation of the grid combined with a scaling by |v| (Supplementary Material), and thereforemust leave the resolution and neuron number invariant. If there is a unique optimal grid, it mustthen also be invariant under this transformation: this constraint is only satisfied by the square grid(v⊥ = 1, v‖ = 0) and the triangular grid (v⊥ =

√3/2, v‖ = 1/2). Between these two, the triangular

grid has the smaller v⊥ and so will minimize neuron number (see Supplementary Material for amore rigorous discussion). We therefore see that the optimality of the triangular lattice is a verygeneral consequence of minimizing neuron number for fixed resolution, and expect the result tohold for a wide range of decoders.

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Figure 4: (A) Our models predict grid scaling ratios that are consistent with experiment. ‘WTA’(Winner-Take-All) and ‘Bayesian’ represent predictions from two decoding models; the dot is thescaling ratio minimizing neuron number and the error bars represent the interval within which theneuron number will be no more than 5% higher than the minimum. For the experimental data, thedot represents the mean measured scale ratio and the error bars represent± one standard deviation.Data were replotted from (8, 13). The dashed red line shows a consensus value running throughthe two theoretical predictions and the two experimental datasets. (B) The mean ratio between gridperiodicity (λi) and the diameter of grid fields (li) in mice (replotted from (14)). Error bars indicate± one S.E.M. For both wild type mice and HCN knockouts (which have larger grid periodicities)the ratio is consistent with

√e (dashed red line). (C) The response lattice of grid cells in rats forms

an equilateral triangular lattice with 60◦ angles between adjacent lattice edges (replotted from (6),n = 45 neurons from 6 rats). Dots represent the outliers.

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Comparison to experiment

Our predictions agree with experiment (8, 13, 14) (see Supplementary Material for details of thedata re-analysis). Specifically, Barry et al., 2007 (Fig. 4A) reported the grid periodicities measuredat three locations along the dorso-ventral axis of of the MEC in rats and found ratios of ∼ 1, ∼ 1.7and ∼ 2.5 ≈ 1.6 × 1.6 relative to the smallest period (13) . The ratios of adjacent scales reportedin (13) had a mean of 1.64± 0.09 (mean ± std. dev., n = 6), which almost precisely matches themean scale factor of

√e predicted from the winner-take-all decoding model, and is also consistent

with the Bayesian decoding model. Recent analysis based on larger data set (8) confirms thegeometric progression of the grid scales. The mean adjacent scale ratio is 1.42±0.17 (mean± std.dev., n = 24) in that data set, accompanied by modest variability of the scaling factors both withinand between animals. These measurements again match both our models (Fig. 4A). The optimalgrid was triangular in both of our models, this again matches measurements (Fig. 4C) (6–8).

The winner-take-all model also predicts the ratio between grid period and grid field width:λi/li = λi/λi+1 =

√e ≈ 1.65. A recent study measured the ratio between grid periodicity and

grid field size to be 1.63± 0.035 (mean ± S.E.M., n = 48) in wild type mice (14), consistent withour predictions (Fig. 4B). This ratio was unchanged, 1.66 ± 0.03 (mean ± S.E.M., n = 86), inHCN1 knockout strains whose absolute grid periodicities increased relative to the wild type (14).The Bayesian model does not make a direct prediction about grid field width; it instead workswith the standard deviation of the posterior P (x | i), σi (Supplementary Material). This parameteris predicted to be σi = 0.19λi in two dimensions, but cannot be directly measured from data. Itis related to the field width li by a proportionality factor whose value depends on detailed tuningcurve shape, noise properties, firing rate, and firing field density (Supplementary Material).

We can estimate the total number of modules, m, by estimating the requisite resolution R2 andusing the relationship m = logR2/ log r2. Assuming that the animal must be able to navigatean environment of area ∼ (10 m)2, with a positional accuracy on the scale of the rat’s body size,∼ (10 cm)2, we get a resolution of R2 ∼ 104. Together with the predicted two-dimensional scalefactor r, this gives m ≈ 10 as an order-of-magnitude estimate. Indeed, in (8), 4-5 modules werediscovered in recordings spanning up to 50% of the dorsoventral extent of MEC; extrapolationgives a total module number consistent with our estimate.

Discussion

We have shown that a grid system with a discrete set of periodicities, as found in the entorhinalcortex, should use a common scale factor r between modules to represent spatial location with thefewest neurons. In one dimension, this organization may be thought of intuitively as implementinga neural analog of a base-b number system. Each scale localizes the animal to some coarse region ofthe environment, and the next scale subdivides that region into b = r “bins” (Fig. 1C). Our problemof minimizing neuron number while fixing resolution is analogous to minimizing the number ofsymbols needed to represent a given range R of numbers in a base-b number system. Specifically,b symbols are required at each of logbR positions, and minimizing the total, b logbR, with respectto b gives an optimal base b = e. Our full theory can thus be seen as a generalization of this simple

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fixed-base representational scheme to noisy neurons encoding two-dimensional location.

The existing data agree with our predictions for the ratios of adjacent scales within the variabil-ity tolerated by our models (Fig. 4). Further tests of our theory are possible. For example, a directgeneralization of our reasoning says that in n-dimensions the optimal ratio between grid scaleswill be near n

√e, with n = 3 having possible relevance to the grid system (15) in, e.g., bats (16).

In general, the theory can be tested by comprehensive population recordings of grid cells alongthe dorso-ventral axis for animals moving in one, two and three dimensional environments. Thereis some evidence that humans also have a grid system (17), in which case our theory may haverelevance to the human sense of place.

We assumed that the grid system should minimize the number of neurons required to achievea given spatial resolution. In fact, any cost which increases monotonically with the number ofneurons would lead to the same optimum. Of course, completely different proposals for the func-tional architecture of the grid system (18,19,23)and associated cost functions will lead to differentpredictions. For example, (18, 19) showed that a grid implementing a “residue number system”(in which adjacent grid scales should be relatively prime) will maximize the range of positionsthat can be encoded. This theory makes distinct predictions for the ratios of adjacent scales (thedifferent periods are relatively prime) and, in its original form, predicts neither the ratio of gridfield width to periodicity nor the organization in higher dimensions, except perhaps by interpretinghigher dimensional grid fields as a product of one-dimensional fields. The essential difference be-tween these two theories lies in the fundamental assumptions: we minimize the number of neuronsneeded to represent space with a given resolution and range, as opposed to maximizing the rangeof locations that may be uniquely encoded.

Grid coding schemes represent position more accurately than place cell codes given a fixednumber of neurons (20, 21). Furthermore, in one dimension a geometric progression of grids thatare self-similar at each scale minimizes the asymptotic error in recovering an animal’s locationgiven a fixed number of neurons (20). The two dimensional grid schemes discussed in this paperwill share the same virtue.

The scheme that we propose may also be more developmentally plausible, as each scale isdetermined by applying a fixed rule (rescaling by r) to the anatomically adjacent scale. This couldbe encoded, for example, by a morphogen with an exponentially decaying concentration gradientalong the dorsoventral axis, something readily attainable in standard models of development. Thisdiffers from the global constraint that all scales be relatively prime, for which the existence of alocal developmental rule is less plausible. As we showed, the location coding scheme that we havedescribed is also robust to variations in the precise value of the scale ratio r, and so would toleratevariability within and between animals.

References and Notes

1. Tolman, E. C. (1948). Cognitive maps in rats and men. Psychol. Rev. 55, 189 .

2. O‘Keefe, J. (1976). Place units in the hippocampus of the freely moving rat. Exp. Neurol. 51, 78 .

3. O‘Keefe, J., Nadel, L. (1978). The Hippocampus as a Cognitive Map (Oxford: Clarendon Press).

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4. Leutgeb, S., Leutgeb, J.K., Barnes, C.A., Moser, E.I., McNaughton, B.L., Moser, M.B. (2005).Independent codes for spatial and episodic memory in the hippocampus. Science, 309, 619.

5. Fyhn, M., Molden, S., Witter, M. P., Moser, E. I., Moser, M. B. (2004). Spatial representation in theentorhinal cortex. Science 305, 1258.

6. Hafting, T., Fyhn, M., Molden, S., Moser, M. B., Moser, E. I. (2005). Microstructure of a spatial mapin the entorhinal cortex. Nature 436, 801.

7. Moser, E. I., Kropff, E., Moser, M. B. (2008). Place cells, grid cells, and the brains spatial representationsystem. Annu. Rev. Neurosci. , 31, 69.

8. Stensola, T. Stensola, T., Solstad T., Froland K., Moser M. B., Moser E. (2012). The entorhinal grid mapis discretized. Nature , 492, 72.

9. Coultrip, R., Granger, R., Lynch, G. (1992). A cortical model of winner-take-all competition via lateralinhibition Neural Networks 5(1), 47-54.

10. Maass, W. (2000). On the computational power of winner-take-all. Neural Comp. 12(11), 2519-2535.

11. de Almeida, L., Idiart, M., Lisman, J.E. (2009). The input-output transformation of the hippocampalgranule cells: from grid cells to place fields. J. Neurosci. 29, 7504.

12. Brun, V.H., Solstad, T., Kjelstrup, K.B., Fyhn, M., Witter, M. P., Moser, E.I., Moser, M.B. (2008).Progressive increase in grid scale from dorsal to ventral medial entorhinal cortex. Hippocampus 18(12),1200.

13. Barry, C., Hayman, R., Burgess, N., Jeffery, K. J. (2007). Experience- dependent rescaling of entorhi-nal grids. Nat. Neurosci. , 10, 682.

14. Giocomo, L.M., Hussaini, S.A., Zheng, F., Kandel, E.R., Moser, M.B., Moser, E.I. (2011). Increasedspatial scale in grid cells of HCN1 knockout mice. Cell 147, 1159.

15. Hayman, R., Verriotis, M.A., Jovalekic, A., Fenton, A.A., Jeffery, K.J. (2011). Anisotropic encodingof three-dimensional space by place cells and grid cells. Nat. Neurosci. , 14, 1182.

16. Yartsev, M. M., Witter,M. P., Ulanovsky, N. (2011). Grid cells without theta oscillations in theentorhinal cortex of bats. Nature, 479, 103.

17. Doeller, C. F., Barry, C., Burgess, N. (2010). Evidence for grid cells in a human memory network.Nature , 463, 657.

18. Fiete, I. R., Burak, Y., Brookings, T. (2008). What grid cells convey about rat location. J. Neurosci. ,28, 6858.

19. Sreenivasan, S., Fiete, I. R. (2011). Grid cells generate an analog error-correcting code for singularlyprecise neural computation. Nat. Neurosci. , 14, 1330.

20. Mathis, A., Herz, A. V. M., Stemmler, M. (2012). Optimal Population Codes for Space: Grid CellsOutperform Place Cells. Neural Computation, 24, 2280-2317.

21. Mathis, A., Herz, A. V. M., Stemmler, M. B. (2012). Resolution of Nested Neuronal RepresentationsCan Be Exponential in the Number of Neurons. Physical Review Letters,109, 018103.

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22. Thue, A. (1892). Om nogle geometrisk taltheoretiske Theoremer. Forandlingerneved de SkandinaviskeNaturforskeres, 14, 352.

23. Stevens C. and Solstad, T., personal communication.

24. NSF grants PHY-1058202, EF-0928048 and PHY-1066293 supported this work, which was completedat the Aspen Center for Physics. VB was also supported by the Fondation Pierre Gilles de Gennes. JPwas supported by the C.V. Starr Foundation. While this paper was being written we became aware ofrelated work in progress by Charles Stevens and Trygve Solstad (personal communication).

13

Supplementary materials

1 Optimizing a “base-b” representation of one-dimensional space

Suppose that we want to resolve location with a precision l in a track of length L. In terms of the resolutionR = L/l, we have argued in the discussion of the main text that a “base-b” hierarchical neural codingscheme will roughly require N = b logbR neurons. To derive the optimal base (i.e. the base that minimizesthe number of the neurons), we evaluate the extremum ∂N/∂b = 0:

∂N/∂b =∂(b logbR)

∂b=∂( b lnR

lnb )

∂b= lnR

ln b− 1

(ln b)2(1)

Setting ∂N/∂b = 0 gives ln b − 1 = 0. Therefore the number of neurons is extremized when b = e. It iseasy to check that this is a minimum.

2 Optimizing the grid system: winner-take-all decoder

2.1 Lagrange multiplier approach

We saw in the main text that, for a winner-take-all decoder, the problem of deriving the optimal ratios of adja-cent grid scales in one dimension is equivalent to minimizing the sum of a set of numbers (N = d

∑mi=1 ri)

while fixing the product (R1 =∏mi=1 ri) to take the fixed value R. Mathematically, it is equivalent to

minimize N while fixing lnR. When N is large we can treat it as a continuous variable and use themethod of Lagrange multipliers as follows. First,we construct the auxiliary function H(r1 · · · rN , β) =N − β (lnR1 − lnR) and then extremize H with respect to each ri and β. Extremizing with respect to rigives

∂H

∂ri= d− β

ri= 0 =⇒ ri =

β

d≡ r . (2)

Next, extremizing with respect to β to implement the constraint on the resolution gives

∂H

∂β= lnR1 − lnR = m ln r − lnR = 0 =⇒ r = R1/m (3)

Having thus implemented the constraint that lnR1 = lnR , it follows that H = N = dmR1/m. Alterna-tively, solving for m in terms of r, we can write H = d r (lnR) / ln r) = d r logr R. It remains to minimizethe number of cells N with respect to r,

∂H

∂r= d lnR

[1

ln r−(

1

ln r

)2]

= 0 =⇒ ln r = 1 (4)

This is in turn implies our resultr = e (5)

for the optimal ratio between adjacent scales in a hierarchical, grid coding scheme for position in onedimension, using a winner-take-all decoder. In this argument we employed the sleight of hand that N andm can be treated as continuous variables, which is approximately valid when N is large. This conditionobtains if the required resolution R is large. A more careful argument is given below that preserves theinteger character of N and m.

14

2.2 Integer N and m

As discussed above, we seek to minimize the sum of a set of numbers (N = d∑m

i=1 ri) while fixing theproduct (R =

∏mi=1 ri) to take a fixed value. We wish to carry out this minimization while recognizing that

the number of neurons is an integer. First, consider the arithmetic mean-geometric mean inequality whichstates that, for a set of non-negative real numbers, x1, x2, ..., xm, the following holds:

(x1 + x2 + ...+ xm)/m ≥ (x1x2...xm)1/m , (6)

with equality if and only if all the xi’s are equal. Applying this inequality, it is easy to see that to minimize∑mi=1 ri, all of the ri should be equal. We denote this common value as r, and we can write r = R1/m.

Therefore, we have

N = dm∑i=1

r = mdR1/m (7)

Suppose R = ez+ε, where z is an integer, and ε ∈ [0, 1). By taking the first derivative of N with respect tom, and setting it to zero, we find that N is minimized when m = z + ε. However, since m is an integer theminimum will be achieved either at m = z or m = z + 1. (Here we used the fact mR1/m is monotonicallyincreasing between 0 and z + ε and is monotonically decreasing between z + ε and∞.) Thus, minimizingN requires either

r = (ez+ε)1z = e

z+εz or r = (ez+ε)

1z+1 = e

z+εz+1 . (8)

In either case, when z is large (and therefore R, N and m are large), r → e. This shows that when theresolution R is sufficiently large, the total number of neurons N is minimized when ri ≈ e for all i.

3 Optimizing the grid system: Bayesian decoder

3.1 Neuron number and resolution

In the main text we argued that the optimal scale factor in one dimension is r = e assuming that decoding isbased on the responses of the most active cell at each scale. However, the decoding strategy could use moreinformation from the population of neurons. Thus, we consider a Bayes-optimal decoder that accounts forall available information by forming a posterior distribution of position, given the activity of all grid cellsin the population. We can make quantitative predictions in this general setting if we assume that the firingof different grid cells is statistically independent and that the tuning curves at each scale i provide dense,uniform, coverage of the interval λi. With these assumptions, the posterior distribution of the animal’sposition, given the activity of grid cells at the single scale i, P (x | i), may be approximated by a series ofGaussian bumps of standard deviation σi spaced at the period λi. Furthermore, σi = cd−1/2li, where li is thewidth of each tuning curve, c is a dimensionless factor incorporating the tuning curve shape and noisiness ofsingle neurons, and d is the coverage factor. The linear dependence on li follows from dimensional analysis.From the definition of d given in the main text, d = ni

liλi

, we see that d can be interpreted as the numberof cells with tuning curves overlapping a given point in space. The square-root dependence of σi on d thenfollows, as this is the effective number of neurons independently encoding position. We assume here that dis large; this is necessary for the Gaussian approximation to hold. Finally, combining the equation for σ withthe relationship, ni = dλili , gives ni = c

√dλiσi . Therefore, the total number of neurons, which we would like

to minimize, is N = c√d∑m

i=1λiσi

.

15

2 6 10 141

2

3

! / "m

ax r* = 2.3

!* = 9.1"

Figure 5: ρmax ≡ maxσ/δ ρ(λσ, σδ) is the scaling factor after optimizing N over σ/δ. The values r∗

and λ∗ are the values chosen by the complete optimization procedure.

In the main text, we minimized N while fixing the resolution R1. In our present Bayesian decodingmodel, R1 will be related to the standard deviation δm of the distribution of location x given the activityof all m scales, Qm(x). In general, the activity of the grid cells at all scales larger than λi provides adistribution over position Qi−1(x) which is combined with the posterior P (x | i) to find the distributionQi(x) given all scales 1 to i. Since we assume independence across scales, Qi−1(x) is obtained by takingthe product over all the posteriors up to scale i− 1: Qi−1(x) = N ∏i−1

j=1 P (x | j), where N normalizes thedistribution. Furthermore, Qi(x) = N ′ P (x | i)Qi−1(x). The posteriors from different scales have differentperiodicities, so multiplying them against each other will tend to suppress all peaks except the central one,which is aligned across scales. We may thus approximate Qi−1(x) and Qi(x) by single Gaussians whosestandard deviations we will denote as δi−1 and δi, respectively. The validity of this approximation is takenup in further detail in section 3.2 below. By dimensional analysis, δi = δi−1/ρ(λiσi ,

σiδi−1

). With the statedGaussianity assumptions, the function ρ may be explicitly defined and evaluated numerically (section 3.2).A Bayes-optimal decoder will then estimate the animal’s position with error proportional to the posteriorstandard deviation over all m scales, δm = (

∏i ρi)

−1 σ1, and no unbiased decoder can do better than this.(We are abbreviating ρi ≡ ρ(λi/σi, σi/δi−1.) Thus, the resolution constraint imposed in the main textbecomes, in the present context, a constraint on

∏i ρi. We will show below that ρ is in fact equal to the scale

factor ri = λi/λi+1.

Thus, we would like to minimize N = c√d∑m

i=1λiσi

subject to a constraint R =∏i ρ(λiσi ,

σiδi−1

). Theminimization is with respect to the parameters λi/σi and σi/δi−1. We perform the calculation in two steps:first optimizing over σi/δi−1, then over λi/σi. The former parameter only affects N indirectly, by changingthe number of scales m through the constraint

∏mi=1 ρ(λiσi ,

σiδi−1

). Choosing σi/δi−1 to maximize ρ willminimize m, and therefore N . We thus replace ρ by ρmax(λ/σ) ≡ maxσ/δ ρ(λ/σ, σ/δ) and minimize Nover the remaining parameters λi/σi. As in the main text, the problem has a symmetry under permutationsof the i, so the optimal λi/σi and σi/δi−1 are independent of i. Thus, m = lnR/ ln ρmax and N ∝

λ/σln ρmax(λ/σ) . We can invert the one-to-one relationship between ρmax and λ/σ (Fig. 5), and minimize Nover ρmax to get ρ∗max = 2.3. In fact, ρ is equal to the scale factor: ρi = ri = λi/λi+1. To see this, expressρi as a product: ρi = δi−1

δi= δi−1

σiσiλi

λiλi+1

λi+1

σi+1

σi+1

δi. Since the factors σi/δi−1 and λi/σi are independent of

i, they cancel in the product and we are left with ρi = λi/λi+1.

16

We have thus seen that the Bayesian decoder predicts an optimal scaling factor r∗ = 2.3 in one dimen-sion. This is similar to, but somewhat different than, the winner-take-all result r∗ = e = 2.7. At a technicallevel the difference arises from the fact that the function ρmax(λ/σ) does not satisfy ρmax = λ

σ as usedpreviously, but is instead more nearly approximated by a linear function with an offset: ρ ≈ α−1(λσ + β). Amore conceptual reason for the difference is that the Gaussian posterior used here has long tails which areabsent in the case with compact firing fields. The scale factor must then be smaller to keep the ambiguoussecondary peaks of the next scale far enough into the tails to be adequately suppressed. The optimizationalso predicts λ∗ = 9.1σ, which may be combined with the formula σ = cd−1/2l to predict l/λ. However,this relationship depends on the parameters c and d which may only be calculated from a more detaileddescription of the single neuron response properties. For this reason, the general Bayesian analysis abovedoes not predict the ratio of the grid periodicity to the width of individual grid fields. Note that λ∗ = 9.1σalso implies that σi/λi+1 ≈ 4 – i.e. that the peaks of the posterior distribution at scale i + 1 are separatedby 4 of the standard deviations of the peaks at scale i.

A similar Bayesian analysis can be carried out for two dimensional grid fields. The posteriors P (x | i)become two-dimensional sums-of-Gaussians, with the centers of the Gaussians laid out on the vertices ofthe grid. Qi(x) is then similarly approximated by a two-dimensional Gaussian. The form of the function ρchanges (section 3.2), but the logic of the above derivation is otherwise unaltered.

3.2 Calculating ρ(λσ ,σδ )

Section 3.1 argued that the function ρ(λσ ,σδ ) can be computed by making the approximation that the posterior

distribution of the animal’s position given the activity at a single scale i, P (x | i), is a periodic sum-of-Gaussians:

P (x | i) =1

2K + 1

K∑n=−K

1√2πσ2

i

e− 1

2σ2i

(x−nλi)2(9)

where K is assumed is large. We further approximate the posterior given the activity of all scales coarserthan λi by a Gaussian with standard deviation δi−1:

Qi−1(x) =1√

2πδ2i−1

e−x2/2δ2i−1 (10)

Assuming independence across scales, it then follows thatQi(x) = P (x | i)Qi−1(x)∫dxP (x | i)Qi−1(x)

. Then ρ(λi/σi, σi/δi−1)

is given by δi−1/δi, where δi is the standard deviation ofQi. We therefore must calculateQi(x) and its vari-ance in order to obtain ρ. After some algebraic manipulation, we find,

Qi(x) =K∑

n=−Kπn

1√2πΣ2

e−(x−µn)2/2Σ2, (11)

where Σ2 =(σ−2i + δ−2

i−1

)−1, µn =(

Σσi

)2λi n, and

πn =1

Ze−n

2λ2i /2(σ2i+δ2i−1). (12)

Z is a normalization factor enforcing∑

n πn = 1. Qi is thus a mixture-of-Gaussians, seemingly contra-dicting our approximation that all the Q are Gaussian. However, if the secondary peaks of P (x | i) are well

17

into the tails of Qi−1(x), then they will be suppressed (quantitatively, if λ2i � σ2

i + δ2i−1, then πn � π0

for |n| ≥ 1), so that our assumed Gaussian form for Q holds to a good approximation. In particular, at thevalues of λ, σ, and δ selected by the optimization procedure described in section 3.1, π1 = 1.3 · 10−3π0. Soour approximation is self-consistent.

Next, we find the variance δ2i :

δ2i = 〈x2〉Qi

=∑n

πn(Σ2 + µ2n)

= Σ2

(1 +

(Σ

σi

)2(λiσi

)2∑n

n2πn

)

= δ2i−1

(1 +

δ2i−1

σ2i

)−1(

1 +

(Σ

σi

)2(λiσi

)2∑n

n2πn

). (13)

We can finally read off ρ(λiσi ,σiδi−1

) as the ratio δi−1/δi:

ρ(λiσi,σiδi−1

) =

(1 +

δ2i−1

σ2i

)1/2(

1 +

(1 +

σ2i

δ2i−1

)−1(λiσi

)2∑n

n2πn

)−1/2

. (14)

For the calculations reported in the text, we took K = 500.

Section 3.1 explained that we are interested in maximizing ρ over σδ , holding λ

σ fixed. The first factorin ρ increases monotonically with decreasing σ

δ ; however,∑

n n2πn also increases and this has the effect of

reducing ρ. The optimal σδ is thus controlled by a tradeoff between these factors. The first factor is relatedto the increasing precision given by narrowing the central peak of P (x | i), while the second factor describesthe ambiguity from multiple peaks.

The derivation can be repeated in the two-dimensional case. We take P (x | i) to be a sum-of-Gaussianswith peaks centered on the vertices of a regular lattice generated by the vectors (λiu, λi~v). We also defineδ2i ≡ 1

2〈|x|2〉Qi . The factor of 1/2 ensures that the variance so defined is measured as an average over thetwo dimensions of space. The derivation is otherwise parallel to the above, and the result is,

ρ2(λiσi,σiδi−1

) =

(1 +

δ2i−1

σ2i

)1/2(

2 +

(1 +

σ2i

δ2i−1

)−1(λiσi

)2∑n,m

|nu+m~v|2 πn,m)−1/2

, (15)

where πn,m = 1Z e−|nu+m~v|2λ2i /2(σ2

i+δ2i−1).

4 Reanalysis of grid data from previous studies

We reanalyzed the data from Barry et. al (13) and Stensola et. al (8) in order to get the mean and the varianceof the ratio of adjacent grid scales. For Barry et. al (13), we first read the raw data from Figure 3b of themain text using the software GraphClick, which allows retrieval of the original (x,y)-coordinates from theimage. This gave the scales of grid cells recorded from 6 different rats. For each animal, we grouped thegrids that had similar periodicities (i.e. differed by less than 20%) and calculated the mean periodicity foreach group. We defined this mean periodicity as the scale of each group. For 4 out of 6 rats, there were 2

18

scales in the data. For 1 out 6 rats, there were 3 grid scales. For the remaining rat, only 1 scale was obtainedas only 1 cell was recorded from that rat. We excluded this rat from further analysis. We then calculated theratio between adjacent grid scales, resulting in 6 ratios from 5 rats. The mean and variance of the ratio were1.64 and 0.09, respectively (n = 6).

For Stensola et. al (8), we first read in the data using GraphClick from Figure 5d of the main text. Thisgave the scale ratios between different grids for 16 different rats. We then pooled all the ratios togetherand calculated the mean and variance. The mean and variance of the ratio were 1.42 and 0.17, respectively(n = 24).

Giocomo et. al (14) reported the ratios between the grid period and the radius of grid field (measured asthe radius of the circle around the center field of the autocorrelation map of the grid cells ) to be 3.26± 0.07and 3.32±0.06 for Wild-type and HCN KO mice, respecitvely. We linearly transform these measurements tothe ratios between grid period and the diameter of the grid field to facilitate the comparison to our theoreticalpredictions. The results are plotted in a bar graph (Fig. 4B in the main text).

Finally, in Figure 4C, we replotted Fig. 1c from (6) by reading in the data using GraphClick and thentranslating that information back into a plot.

5 General optimality of the triangular lattice

Our task is to minimize the number of neurons in a population made up ofmmodules,N = d∑m

i=1 |v⊥|(λili )2,subject to a constraint on resolution R = F ({λ, l,u,v},m). The specific form of the resolution function Fwill, of course, depend on the details of tuning curve shape, noise, and decoder performance. Nevertheless,we will prove that the triangular lattice is optimal in all models sharing the following general properties:

• Uniqueness: Our optimization problem has a unique solution for all R. The optimal parameters arecontinuous functions of R.

• Symmetry: Simultaneous rotation of all firing rate maps leaves F invariant. Likewise, F is invariantunder simultaneous rescaling of all maps. These transformations are manifestly symmetries of theneuron numberN . Rotation invariance implies that F depends on u and v only through the two scalarparameters v⊥ and v‖ (the components of v orthogonal to and parallel to u, respectively). Scale in-variance implies that the dependence on the dimensionful parameters {λ, l} is only through the ratios{r, λ/l}, where ri = λi/λi+1 are the scale factors. The resolution formulas in both the winner-take-all and the Bayesian formulations are evidently scale-invariant, as they depend only on dimensionlessratios of grid parameters. We will also assume that firing fields are circularly-symmetric.

• Asymptotics: The resolution F ({r, λ/l}, v‖, v⊥,m) increases monotonically with each λi/li. Whenall λi/li → ∞, the grid cells are effectively place cells and so the grid geometry cannot matter.Therefore, F becomes independent of v in this limit.

We will first argue that the uniqueness and symmetry properties imply that the optimal lattice can onlybe square or triangular. The asymptotic condition then picks out the triangular grid as the better of these two.To see the implications of the symmetry condition, consider the following transformation of the parameters:

v⊥ → −v⊥/|v|2

v‖ → v‖/|v|2

li → li/|v|

19

This takes the vector v, reflects it through u (keeping the same angle with u), and scales it to have length1/|v|. This new v, together with u, thus generates the same lattice as the original u and v, but rotated,scaled, and with the roles of u and v exchanged. We then also scale all field width parameters by thesame factor 1/|v| to compensate for the stretching of the lattice. And although this is a rotation of thelattice and not the firing fields, our assumed isotropy of the firing fields implies that the transformation isindistinguishable from a rotation of the entire rate map. Since the overall transformation is equivalent to acommon rotation and scaling of all rate maps, it will (by our symmetry assumption) leave the neuron numberand resolution unchanged. If the optimal lattice is unique, it must then be invariant under this transformation.

Which lattices are invariant under the above transformation? It must take the generator v to anothergenerator v′ of the same lattice. This requirement demands that the generators are related by a modulartransformation:

v′ = av + bu

u = cv + du,

with a, b, c, d integers such that |ad − bc| = 1. The second equation, and linear independence of u and v,require c = 0, d = 1 and so |a| = 1. Plugging in our transformation of v, the first equation then givesa = −1, |v| = 1 and v‖ = b/2. Since v + nu will generate the same lattice as v, for any integer n, we mayassume 0 ≤ v‖ < 1. The only solutions are the square lattice with v‖ = 0, v⊥ = 1 and the triangular latticewith v‖ = 1/2, v⊥ =

√3/2.

It remains to choose between these two possibilities. We want to minimize N = d∑

i |v⊥|(λili )2, so itseems that we should minimize |v⊥|, giving the triangular lattice. However, the constraint on resolution willintroduce v−dependence into λ/l, so it is not immediately clear that we can minimize N by minimizing|v⊥| alone. But the asymptotic condition implies the existence of a large-R regime tied to large λ/l, andasserts that in this limit the v-dependence drops out. Therefore, the triangular lattice is optimal for largeenough R. Since the only other possible optimum is the square lattice, and our uniqueness assumptionprevents the solution from changing discontinuously as R is lowered, it must be the case that the triangularlattice is optimal for all R.

20